Kolmogorov extension theorem
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In mathematics, the Kolmogorov extension theorem is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a stochastic process. It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov.
[edit] Statement of the theorem
Let T denote some interval (thought of as "time"), and let . For each and finite sequence of times , let be a probability measure on . Suppose that these measures satisfy two consistency conditions:
- for all permutations π of and measurable sets ,
- for all measurable sets ,
Then there exists a probability space and a stochastic process such that
for all , and measurable sets , i.e. X has the as its finite-dimensional distributions.
[edit] Reference
- Øksendal, Bernt (2003). Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin. ISBN 3-540-04758-1.